Spectral Bellman Method: Unifying Representation and Exploration in RL

2025-07-17

AI Summary

Existing representation learning methods lack structural alignment with value-based reinforcement learning—particularly Bellman updates—leading to a misalignment between representation learning and policy optimization. Method: We propose the Spectral Bellman Representation framework, which theoretically establishes, for the first time, an intrinsic spectral relationship between the value function distribution under the Bellman operator and the feature covariance matrix. Leveraging multi-step Bellman operators and spectral analysis, we derive an optimization objective strictly aligned with Bellman dynamics and introduce the Inherent Bellman Error constraint to jointly govern representation learning and exploration. Contribution/Results: Our method requires only lightweight modifications to existing algorithms yet achieves significant performance gains on challenging exploration and long-horizon credit assignment tasks. Empirical results demonstrate its effectiveness, generalization capability, and structural soundness, validating the theoretical foundation and practical utility of spectral alignment in representation learning for RL.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

The effect of representation has been demonstrated in reinforcement learning, from both theoretical and empirical successes. However, the existing representation learning mainly induced from model learning aspects, misaligning with our RL tasks. This work introduces Spectral Bellman Representation, a novel framework derived from the Inherent Bellman Error (IBE) condition, which aligns with the fundamental structure of Bellman updates across a space of possible value functions, therefore, directly towards value-based RL. Our key insight is the discovery of a fundamental spectral relationship: under the zero-IBE condition, the transformation of a distribution of value functions by the Bellman operator is intrinsically linked to the feature covariance structure. This spectral connection yields a new, theoretically-grounded objective for learning state-action features that inherently capture this Bellman-aligned covariance. Our method requires a simple modification to existing algorithms. We demonstrate that our learned representations enable structured exploration, by aligning feature covariance with Bellman dynamics, and improve overall performance, particularly in challenging hard-exploration and long-horizon credit assignment tasks. Our framework naturally extends to powerful multi-step Bellman operators, further broadening its impact. Spectral Bellman Representation offers a principled and effective path toward learning more powerful and structurally sound representations for value-based reinforcement learning.

Problem

• Aligns representation learning with Bellman updates in RL
• Discovers spectral link between value functions and feature covariance
• Improves exploration and performance in hard RL tasks

Innovation

• Spectral Bellman Representation aligns with Bellman updates
• Feature covariance linked to Bellman operator transformation
• Simple modification enables structured exploration in RL